The hypervolume indicator has frequently been used for comparing evolutionary multi-objective optimization (EMO) algorithms. A reference point is needed for hypervolume calculation. However, its specification has not been discussed in detail from a viewpoint of fair performance comparison. A slightly worse point than the nadir point is usually used for hypervolume calculation in the EMO community. In this paper, we propose a reference point specification method for fair performance comparison of EMO algorithms. First, we discuss the relation between the reference point specification and the optimal distribution of solutions for hypervolume maximization. It is demonstrated that the optimal distribution of solutions strongly depends on the location of the reference point when a multi-objective problem has an inverted triangular Pareto front. Next, we propose a reference point specification method based on theoretical discussions on the optimal distribution of solutions. The basic idea is to specify the reference point so that a set of well-distributed solutions over the entire linear Pareto front has a large hypervolume and all solutions in such a solution set have similar hypervolume contributions. Then, we examine whether the proposed method can appropriately specify the reference point through computational experiments on various test problems. Finally, we examine the usefulness of the proposed method in a hypervolume-based EMO algorithm. Our discussions and experimental results clearly show that a slightly worse point than the nadir point is not always appropriate for performance comparison of EMO algorithms.

中文翻译：

---

如何在 Hypervolume 计算中指定参考点以进行公平的性能比较

hypervolume 指标经常用于比较进化多目标优化 (EMO) 算法。超体积计算需要一个参考点。但是，从公平的性能比较的角度来看，尚未详细讨论其规范。比最低点稍差的点通常用于 EMO 社区中的超容量计算。在本文中，我们提出了一种用于 EMO 算法公平性能比较的参考点规范方法。首先，我们讨论参考点规范与超体积最大化解的最优分布之间的关系。结果表明，当多目标问题具有倒三角帕累托前沿时，解的最优分布强烈依赖于参考点的位置。下一个，我们提出了一种基于对解决方案最优分布的理论讨论的参考点规范方法。基本思想是指定参考点，以便在整个线性帕累托前沿上的一组分布良好的解决方案具有较大的超体积，并且此类解决方案集中的所有解决方案都具有相似的超体积贡献。然后，我们通过对各种测试问题的计算实验来检查所提出的方法是否可以适当地指定参考点。最后，我们检查了所提出的方法在基于超体积的 EMO 算法中的有用性。我们的讨论和实验结果清楚地表明，比最低点稍差的点并不总是适合 EMO 算法的性能比较。